If we are running monthly cross$-$sectional regressions where we regress excess stock returns on stock characteristics such as market capitalization, book$-$to$-$market ratio $($B$/$M$),$ plus a new factor like the "Twitter factor" for all companies in a given month, and then we extract monthly betas for the Twitter factor $($Cross$-$sectional regression$),$ and regress those betas on Fama$-$French factors $($Time series regression $)$and obtain a significant alpha, this indicates an additional premium associated with the Twitter factor not explained by the existing factors.

To analyze if this Twitter factor is compensation for some specific risk, such as default risk represented by the Altman Z$-$score, what approach should we use?

Approach $1$: Use the Altman Z$-$score for each company for each month as a control variable in the initial cross$-$sectional regression, then fetch monthly betas for the Twitter factor and regress those betas on the existing Fama$-$French factors in time series regression to see if the alpha is still significant.

Approach $2$: Use the Altman Z$-$score for each company for each month as a control variable in the initial cross$-$sectional regression, then fetch monthly betas for the Twitter factor and regress those betas on the existing Fama$-$French factors in time series factor, but also include an aggregate Altman Z$-$score for all firms in the time$-$series regression to see if the alpha is still significant.

Which approach is more robust for determining whether the Twitter factor is compensation for default risk, and why? Our goal is to find a similar factor as Fama and French found.